Map projection

A map projection is any of many methods used in cartography (mapmaking) to represent the two-dimensional curved surface of the earth or other body on a plane. The term "projection" here refers to any function defined on the earth's surface and with values on the plane, and not necessarily a geometric projection.

Flat maps could not exist without map projections. Flat maps can be more useful than globes in many situations: they are more compact and easier to store; they readily accommodate an enormous range of scales; they are viewed easily on computer displays; they can facilitate measuring properties of the terrain being mapped; they can show larger portions of the earth's surface at once; and they are cheaper to produce and transport. These useful traits of flat maps motivate the development of map projections.

Metric properties of maps

Many properties can be measured on the earth's surface independently of its geography. Some of these properties are

• area
• shape
• direction
• bearing
• distance
• scale

Map projections can be constructed to preserve one or some of these properties, though not all of them simultaneously. Each projection preserves or compromises or approximates basic metric properties in different ways. The purpose of the map, then, determines which projection should form the base for the map. Since many purposes exist for maps, so do many projections exist upon which to construct them.

Another major concern that drives the choice of a projection is the compatibility of data sets. Data sets are geographic information. As such, their collection depends on the chosen model of the earth. Different models assign slightly different coordinates to the same location, so it is important that the model be known and that chosen projection is compatible with that model. On small areas (large scale) data compatibility issues are more important since metric distortions are minimal at this level. In very large areas (small scale), on the other hand, distortion is a more important factor to consider.

Construction of a map projection

The creation of a map projection involves three steps.

1. selection of a model for the shape of the earth or planetary body (usually choosing between a sphere or ellipsoid).
2. transform geographic coordinates (longitude and latitude) to plane coordinates (eastings and northings or x,y).
3. reduce the scale (it does not matter in what order the second and third steps are performed).

Because the real earth's shape is irregular, information is lost in the first step, in which an approximating, regular model is chosen. Reducing the scale may be considered to be part of transforming geographic coordinates to plane coordinates.

Most map projections, both practically and theoretically, are not "projections" in any physical sense. Rather, they depend on mathematical formulae that have no direct physical interpretation. However, in understanding the concept of a map projection it is helpful to think of a globe with a light source placed at some definite point with respect to it, projecting features of the globe onto a surface. The following discussion of developable surfaces is based on that concept.

Choosing a projection surface

If a surface can be transformed onto another surface without stretching, tearing, or shrinking, then the surface is said to be an applicable surface. The sphere and ellipsoid are not applicable with a plane surface, so any projection that attempts to project them on a flat sheet will have to distort the image (similar to the impossibility of making a flat sheet from an orange peel). A surface that can be unfolded or unrolled into a flat plane or sheet without stretching, tearing or shrinking is called a 'developable surface'. The cylinder, cone and of course the plane are all developable surfaces.

One way of describing a projection is describing a projection from the earth's surface to a cylinder or cone. Together with the simple second step of transforming the latter two into a plane we have the full projection. The first step inevitably distorts some properties of the globe, the developable surface may be unfolded without further distortion.

Orientation of the projection

Once a choice is made between projecting onto a cylinder, cone, or plane, the orientation of the shape must be chosen. The orientation is how the shape is placed with respect to the globe. The orientation of the projection surface can be normal (inline with the earth's axis), transverse (at right angles to the earth's axis) or oblique (any angle in between). These surfaces may also be either tangent or secant to the spherical or ellipsoidal globe. Tangent means the surface touches but does not slice through the globe; secant means the surface does slice through the globe. Insofar as preserving metric properties go, it is never advantageous to move the developable surface away from contact with the globe, so that practice is not discussed here.

Scale

A globe is the only way to represent the earth with the same scale throughout the entire map surface and in all directions. For a flat map this is not even possible across an area of any extent.

Thus, on a flat map, properties of constant scale are always limited.

Possible properties are:

• the scale depends on location, but not on direction; this is equivalent with preservation of angles: conformal map
• for a given latitude and direction, the scale is the same everywhere; this applies for any cylindrical projection
• combination of the two: the scale depends on latitude only, not on longitude or direction; this applies for the Mercator projection

Choosing a model for the shape of the Earth

Projection construction is also affected by how the shape of the earth is approximated. In the following discussion on projection categories, a sphere is assumed. However, the Earth is not exactly spherical but is closer in shape to an oblate ellipsoid, a shape which bulges around the equator. Selecting a model for a shape of the earth involves choosing between the advantages and disadvantages of a sphere versus an ellipsoid. Spherical models are useful for small-scale maps such as world atlases and globes, since the error at that scale is not usually noticeable or important enough to justify using the more complicated ellipsoid. The ellipsoidal model is commonly used to construct topographic maps and for other large and medium scale maps that need to accurately depict the land surface.

A third model of the shape of the earth is called a geoid, which is a complex and more or less accurate representation of the global mean sea level surface that is obtained through a combination of terrestrial and satellite gravity measurements. This model is not used for mapping due to its complexity but is instead used for control purposes in the construction of geographic datums. (In geodesy, plural of "datum" is "datums," rather than "data".) A geoid is used to construct a datum by adding irregularities to the ellipsoid in order to better match the Earth's actual shape (it takes into account the large scale features in the Earth's gravity field associated with mantle convection patterns, as well as the gravity signatures of very large geomorphic features such as mountain ranges, plateaus and plains). Historically, datums have been based on ellipsoids that best represent the geoid within the region the datum is intended to map. Each ellipsoid has a distinct major and minor axis. Different controls (modifications) are added to the ellipsoid in order to construct the datum, which is specialized for a specific geographic regions (such as the North American Datum). A few modern datums, such as the one used in the Global Positioning System GPS, are optimized to represent the entire earth as well as possible with a single ellipsoid, at the expense of some accuracy in smaller regions.

Categories

A fundamental projection classification is based on type of projection surface onto which the globe is conceptually projected. The projections are described in terms of placing a gigantic surface in contact with the earth, followed by an implied scaling operation. These surfaces are cylindrical (e.g., Mercator), conic (e.g., Albers), and azimuthal or plane (e.g., stereographic). Many mathematical projections, however, do not neatly fit into any of these three conceptual projection methods. Hence other peer categories have been described in the literature, such as pseudoconic, pseudocylindrical, pseudoazimuthal, retroazimuthal, and polyconic.

Another way to classify projections is through the properties they retain. Some of the more common categories are

• direction preserving, called azimuthal (but only possible from the central point)
• locally shape-preserving, called conformal or orthomorphic
• area-preserving, called equal-area or equiareal or equivalent or authalic
• distance preserving - equidistant (preserving distances between one or two points and every other point)
• shortest-route preserving - gnomonic projection

NOTE: It is impossible to construct a map projection that is both equal-area and conformal.

Organized by surface

Cylindrical

The term "cylindrical projection" is used to refer to any projection in which meridians are mapped to equally spaced vertical lines and circles of latitude (parallels) are mapped to horizontal lines (or, mutatis mutandis, more generally, radial lines from a fixed point are mapped to equally spaced parallel lines and concentric circles around it are mapped to perpendicular lines).

The mapping of meridians to vertical lines can be visualized by imagining a cylinder (of which the axis coincides with the Earth's axis of rotation) wrapped around the Earth and then projecting onto the cylinder, and subsequently unfolding the cylinder.

Unavoidably there is an east-west stretching away from the equator by a factor equal to the secant of the latitude, compared with the scale at the equator.

The various cylindrical projections can be described in terms of whether this stretching is:

• accompanied by a corresponding north-south stretching, so that at each location the east-west scale is the same as the north-south-scale: conformal cylindrical or Mercator; with regard to area this distorts even more (see also transverse Mercator).
• compensated for by a corresponding north-south compression: equal-area cylindrical (with many named subordinates such as Gall-Peters or Gall orthographic, Behrmann, and Lambert cylindrical equal-area); this corresponds to horizontal projection from the Earth onto the cylinder: this divides north-south distances by a factor equal to the secant of the latitude, cancelling the other secant factor with regard to area, but distorting squares to rectangles even more, with a scale ratio of the secant squared
• not accompanied by a north-south stretching or compression: equidistant cylindrical or plate carrée.
• Miller cylindrical.
• geometric projection from the center (central cylindrical projection); not very suitable, because the north-south stretching is by a factor of the secant squared, even more than in the Mercator projection, and thus excessive

In the cases other than the first, the latitude where the east-west scale is the same as the north-south-scale may be the equator or elsewhere; in the first case everywhere away from the equator the east-west scale is larger than the north-south-scale, in the second case for part of the map larger and for part of the map smaller.

A map of the whole Earth is a rectangle, except in the first case, where it is a vertically infinite strip of constant width.

Pseudocylindrical

Pseudocylindrical projections represent the central meridian and each parallel as a straight line, but not the other meridians. Each pseudocylindrical projection represents a point on the Earth along the straight line representing its parallel, at a distance which is a function of its difference in longitude from the central meridian.

• sinusoidal: the north-south scale is the same everywhere at the central meridian, and the east-west scale is throughout the map the same as that; correspondingly, on the map, like in reality, the length of each parallel is proportional to the cosine of the latitude; thus the shape of the map for the whole earth is the area between two symmetric rotated cosine curves [1] (http://mathworld.wolfram.com/SinusoidalProjection.html); the true distance between two points on the same meridian corresponds to the distance on the map between the two parallels, which is smaller than the distance between the two points on the map; the meridians drawn on the map help the user realizing the distortion and mentally compensating for it
• Mollweide
• Goode's homolosine
• Eckert IV and Eckert VI

Conic

• equidistant conic
• Lambert conformal conic
• Albers conic

Pseudoconic

• Bonne
• Werner cordiform designates a pole and a meridian; distances from the pole are preserved, as are distances from the meridian (which is straight) along the parallels.
• continuous American polyconic

Azimuthal

Azimuthal projections have the property that directions from a central point are preserved (and hence, great circles through the central point are represented by straight lines on the map). Usually these projections also have radial symmetry in the scales and hence in the distortions: map distances from the central point are computed by a function r(d) of the true distance d, independent of the angle; correspondingly, circles with the central point as center are mapped into circles which have as center the central point on the map..

The mapping of radial lines can be visualized by imagining a plane tangent to the Earth, with the central point as tangent point.

The radial scale is r'(d) and the transverse scale r(d)/(R sin (d/R)) where R is the radius of the Earth.

Some azimuthal projections are true perspective projections; that is, they can be constructed mechanically, projecting the surface of the Earth by extending lines from a points of perspective (along an infinite line through the tangent point and the tangent point's antipode) onto the plane.

• azimuthal equidistant: r(d) = cd; it is used by amateur radio operators to know the direction to point their antennas toward a point and see the distance to it. Distance from the tangent point on the map is proportional to surface distance on the earth ([2] (http://mathworld.wolfram.com/AzimuthalEquidistantProjection.html); for the case where the tangent point is the North Pole, see the flag of the United Nations)
• Lambert azimuthal equal-area. Distance from the tangent point on the map is proportional to straight-line distance through the earth: r(d) = c sin (d/2R) [3] (http://mathworld.wolfram.com/LambertAzimuthalEqual-AreaProjection.html)
• azimuthal conformal projection is the same as stereographic. It can be constructed by using the tangent point's antipode as the point of perspective. r(d) = c tan (d/2R); the scale is c/(2R cos²(d/2R))[4] (http://mathworld.wolfram.com/StereographicProjection.html)
• orthographic maps each point on the earth to the closest point on the plane. Can be constructed from a point of perspective an infinite distance from the tangent point; r(d) = c sin (d/R) [5] (http://mathworld.wolfram.com/OrthographicProjection.html)
• gnomonic displays great circles as straight lines. Can be constructed by using a point of perspective at the center of the Earth. r(d) = c tan (d/R); a hemisphere already requires an infinite map [6] (http://mathworld.wolfram.com/GnomonicProjection.html), [7] (http://members.shaw.ca/quadibloc/maps/maz0201.htm)
• logarithmic azimuthal is constructed so that each point's distance from the center of the map is the logarithm of its distance from the tangent point on the Earth. Works well with cognitive maps. r(d) = c ln (d/d₀); locations closer than at a distance equal to the constant d₀ are not shown ([8] (http://www.gis.psu.edu/projection/chap6figs.html), figure 6-5)

Organized by preservation of a metric property

Conformal

Conformal map projections preserve angles locally.

• Mercator
• stereographic
• Lambert conformal conic

Equal-area

These projections preserve area.

• Gall orthographic (also known as Peters projection)
• Albers conic
• Lambert azimuthal equal-area
• Mollweide
• Briesemeister
• sinusoidal
• Goode's homolosine
• Hobo-Dyer

Equidistant

These preserve distance from some standard point or line.

• Plate carrée - north-south scale is constant
• Equirectangular - equal distance between all latitudes and longitudes.
• azimuthal equidistant - radial scale with respect to the central point is constant
• equidistant conic
• sinusoidal - east-west scale is constant and corresponds to distances between parallels (but the north-south scale away from the central meridian is larger due to the obliqueness of the meridians)
• Werner cordiform distances from the North Pole are correct as are the curved distance on parallels
• Soldner
• two-point equidistant

Gnomonic

Great circles are displayed as straight lines:

• gnomonic projection

Retroazimuthal

Direction to a fixed location B (the bearing at the starting location A of the shortest route) corresponds to the direction on the map from A to B.

• Hammer retroazimuthal - also preserves distance from the central point
• Craig retroazimuthal aka Mecca or Qibla - also has vertical meridians

Compromise projections

Compromise projections give up the idea of perfectly preserving metric properties, seeking instead to strike a balance between distortions, or to simply make things "look right".

• Robinson
• Van der Grinten
• Miller cylindrical
• Winkel tripel

Other noteworthy projections

• Buckminster Fuller's Dymaxion
• Chamberlin trimetric

References

• Fran Evanisko, American River College, lectures for Geography 20: "Cartographic Design for GIS", Fall 2002
• Snyder, J.P., Album of Map Projections, United States Geological Survey Professional Paper 1453, United States Government Printing Office, 1989.
• Synder, J.P., Map Projections - A Working Manual, United States Geological Survey Professional Paper 1395, United States Government Printing Office, 1987.

See also: Cartographer, GIS

External Links

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• US Geological Survey overview (http://erg.usgs.gov/isb/pubs/MapProjections/projections.html)
• Map projections intro (http://members.shaw.ca/quadibloc/maps/mapint.htm)
• Mathworld formulae (http://mathworld.wolfram.com/topics/MapProjections.html)
• How Projections Work (http://www.progonos.com/furuti/MapProj/Normal/CartHow/cartHow.html)
• PDFs of projections (http://www.ilstu.edu/microcam/map_projections/)
• GIFs of projections (http://www.mapthematics.com/Projections.html)
• Java applet for interactive projections (http://www.btinternet.com/~se16/js/mapproj.htm)
• USGS info (http://www.3dsoftware.com/Cartography/USGS/MapProjections/)
• Geodesy, Cartography and Map Reading from Colorado State University (http://www.cnr.colostate.edu/class_info/nr502/mainpage/course_mainpage.html)
• A collection of map projections and reference systems for Europe (http://www.mapref.org/)
• What is a map projection? (http://kartoweb.itc.nl/geometrics/Map%20projections/body.htm)

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